Chaotic vibrations of beams on nonlinear elastic foundations subjected to reciprocating loads

Chaotic vibration of beams resting on a foundation with nonlinear stiffness is investigated in this paper. Cosine–cosine function is employed in modeling of the reciprocating load. The equation of motion is derived and solved to obtain corresponding Poincaré section in phase–space. Lyapunov exponent as a criterion for chaos indication is obtained. Dynamic behavior of the beam is examined in resonance condition. Homoclinic orbits are captured and their corresponding Melnikov's functions are established. A parametric study is then carried out and effects of linear and nonlinear parameters on the chaotic behavior of the system are studied.     

Dynamic response of beams on generalised elastic foundations

Dynamic responses of beams on generalised elastic foundations is studied using the method of Initial parameters. The foundation model proposed by Vlasov and Leontev is modified by incorporating in the analysis the horizontal displacements in the elastic foundation thus making it more general and physically close to the actual situation. Results are compared with those reported by Rades, using Pasternak's foundation model and Winkler's model. The insufficiency of the Winkler's model in the study of dynamic responses (mainly the bending moments) is emphasized. Solutions presented are quite general for application to beams on generalised elastic foundations subjected to arbitrary external dynamic loads and (or) moments.     

弹性地基上Timoshenko梁的精确数值解

研究了弹性地基上Timoshenko梁的高精度有限元分析方法,利用控制微分方程的基本解建立了单元形函数,提出了弹性地基上Timoshenko梁分析的Trefftz单元。通过对引入的非节点自由度进行静力凝聚,得到的精确单元与常规单元具有相同的节点自由度。文中还讨论了有效降低计算过程中舍入误差的方法。算例结果表明,采用提出...     

Generalized solutions of beams with jump discontinuities on elastic foundations

Summary  The bending solutions of the Euler–Bernoulli and the Timoshenko beams with material and geometric discontinuities are developed in the space of generalized functions. Unlike the classical solutions of discontinuous beams, which are expressed in terms of multiple expressions that are valid in different regions of the beam, the generalized solutions are expressed in terms of a single expression on the entire domain. It is shown that the boundary-value problems describing the bending of beams with jump discontinuities on discontinuous elastic foundations have more compact forms in the space of generalized functions than they do in the space of classical functions. Also, fewer continuity conditions need to be satisfied if the problem is formulated in the space of generalized functions. It is demonstrated that using the theory of distributions (i.e. generalized functions) makes finding analytical solutions for this class of problems more efficient compared to the traditional methods, and, in some cases, the theory of distributions can lead to interesting qualitative results. Examples are presented to show the efficiency of using the theory of generalized functions. Received 6 June 2000; accepted for publication 24 October 2000     

Structural modeling of beams on elastic foundations with elasticity couplings

A new simplified structural model and its governing equations for beams on elastic foundations with elastic coupling are proposed. This modeling system is simple but appropriate for the initial structural design of large-scale submerged floating-beam structures moored by tension legs spaced at uniform interval along the beam. The model is actually for beam on discrete elastic supports rather than on continuous elastic foundations. Therefore, the governing equations are based on finite difference calculus and solutions for beams on discrete elastic supports with elasticity coupling are also proposed. To clarify the applicability limit of the proposed model, the equivalence between a beam on discrete elastic supports and that on continuous elastic foundation is investigated by comparisons of characteristic solutions.     

Winkler-Pasternak弹性地基梁自由振动的二维弹性分析

基于二维线弹性理论,应用Hamilton原理,获得Winkler-Pasternak弹性地基梁自由振动的控制微分方程,应用微分求积法(DQM)数值研究了梁自由振动的无量纲频率特性。计算结果与已有的结果(Bernoulli-Euler梁和Timoshenko梁)比较表明,本文的分析方法对弹性地基长梁和短梁自由振动的研究都有效。最后考虑了几何参数对梁频率的影响,以及不同边界条件下地基系数对频率的影响和收敛性。     

Vibrations and transient responses of Timoshenko beams resting on elastic foundations

Summary A finite element technique is used to obtain the natural frequencies and transient responses of Timoshenko beams resting on elastic foundations. The beam is discretized into beam elements, each with four degrees of freedom. The equations of motion in terms of the nodal degrees of freedom are derived by applying Hamilton's principle. The numerical results for a hinged-hinged beam are given to show the effects of rotatory inertia, shear deformation and foundation constants on the natural frequencies of the beam. The transient responses of beams are subsequently presented and compared with the available solutions.

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Eigenschwingungen und transientes Verhalten von Timoshenko-Balken mit elastischer Bettung

Übersicht Mit einer Finit-Element-Technik werden die Eigenfrequenzen und das transiente Verhalten von Timoshenko-Balken mit elastischer Bettung ermittelt. Der Balken wird dabei in Elemente mit je vier Freiheitsgraden diskretisiert. Die Bewegungsgleichungen werden mit Hilfe des Prinzips von Hamilton hergeleitet. Numerische Ergebnisse für einen beidseitig gelenkig gelagerten Balken zeigen den Einfluß von Drehträgheit, Schubverformung und Bettungskonstanten auf die Eigenfrequenzen. Anschließend wird das transiente Verhalten dargestellt und mit bekannten Ergebnissen verglichen.

     

Vibration of thermally postbuckled carbon nanotube-reinforced composite beams resting on elastic foundations

This paper investigates the small- and large-amplitude vibrations of thermally postbuckled carbon nanotube-reinforced composite (CNTRC) beams resting on elastic foundations. For the CNTRC beams, uniformly distributed (UD) and functionally graded (FG) reinforcements are considered where the temperature-dependent material properties of CNRTC beams are assumed to be graded in the thickness direction and estimated through a micromechanical model. The motion equations are derived based on a higher order shear deformation beam theory with including the beam-foundation interaction. The initial deflection caused by thermal postbuckling is also included. The numerical illustrations concern small- and large-amplitude vibration characteristics of thermally postbuckled CNTRC beams under uniform temperature field. The effects of carbon nanotube (CNT) volume fraction and distribution patterns as well as foundation stiffness on the vibration characteristics of CNTRC beams are examined in detail.     

弹性地基上转动FGM梁自由振动的DTM分析

基于Euler-Bernoulli梁理论,利用广义Hamilton原理推导得到弹性地基上转动功能梯度材料(FGM)梁横向自由振动的运动控制微分方程并进行无量纲化,采用微分变换法(DTM)对无量纲控制微分方程及其边界条件进行变换,计算了弹性地基上转动FGM梁在夹紧-夹紧、夹紧-简支和夹紧-自由三种边界条件下横向自由振动的无量纲固有频率,再将控制微分方程退化到无转动和地基时的FGM梁,计算其不同梯度指数时第一阶无量纲固有频率值,并和已有文献的FEM和Lagrange乘子法计算结果进行比较,数值完全吻合。计算结果表明,三种边界条件下FGM梁的无量纲固有频率随无量纲转速和无量纲弹性地基模量的增大而增大;在一定无量纲转速和无量纲弹性地基模量下,FGM梁的无量纲固有频率随着FGM梯度指数的增大而减小;但在夹紧-简支和夹紧-自由边界条件下,一阶无量纲固有频率几乎不变。     

CC series solution for bending of rectangular plates on elastic foundations

The analytical solution for the bending problem of the rectangular plates on an elastic foundation is investigated by using the Stockes' transformation of a double variables function. The numerical results for the rectangular plates with free edges on the elastic foundations under a concentrated force are given in the example. First Received Dec. 14 1993     

Effect of transverse shears on complex nonlinear vibrations of elastic beams

Models of geometrically nonlinear Euler-Bernoulli, Timoshenko, and Sheremet’ev-Pelekh beams under alternating transverse loading were constructed using the variational principle and the hypothesis method. The obtained differential equation systems were analyzed based on nonlinear dynamics and the qualitative theory of differential equations with using the finite difference method with the approximation O(h²) and the Bubnov-Galerkin finite element method. It is shown that for a relative thickness λ ⩽ 50, accounting for the rotation and bending of the beam normal leads to a significant change in the beam vibration modes.     

Subharmonic and ultra-subharmonic response of nonlinear elastic beams subjected to harmonic excitation

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Nonlinear vibration of functionally graded graphene-reinforced composite laminated beams resting on elastic foundations in thermal environments

Modeling and nonlinear vibration analysis of graphene-reinforced composite (GRC) laminated beams resting on elastic foundations in thermal environments are presented. The graphene reinforcements are assumed to be aligned and are distributed either uniformly or functionally graded of piece-wise type along the thickness of the beam. The motion equations of the beams are based on a higher-order shear deformation beam theory and von Kármán strain displacement relationships. The beam–foundation interaction and thermal effects are also included. The temperature-dependent material properties of GRCs are estimated through a micromechanical model. A two-step perturbation approach is employed to determine the nonlinear-to-linear frequency ratios of GRC laminated beams. Detailed parametric studies are carried out to investigate the effects of material property gradient, temperature variation, stacking sequence as well as the foundation stiffness on the linear and nonlinear vibration characteristics of the GRC laminated beams.     

A certain approach to the solution of boundary value problems for the theory of plates and beams

Institute of Mechanics, Academy of Sciences of the Ukrainian SSR, Kiev. Translated from Prikladnaya Mekhanika, Vol. 25, No. 9, pp. 96–101, September, 1989.     

非线性弹性大变形材料梁的抗爆特性

从理论上探讨了非线性弹性大变形材料应用于抗爆结构的可行性,为此,基于等效结构体系的分析原理,将两端固定铰支梁的横向和纵向位移表示为三角级数形式,应用第二类Lagrange方程建立了非线性大变形材料梁的非线性分析方法,并且用ABAQUS有限元软件中的超弹性材料模型验证了所提出的方法的有效性。对典型的爆炸荷载作用下非线性弹性大变形材料梁的抗爆特性进行了分析,讨论了动力放大系数和材料性质及动荷载之间的关系。结果表明:与线弹性小变形材料相比,非线性弹性大变形材料具有优良的抗爆特性,结构的抗爆能力随结构变形的增大而显著提高。     

Shear stresses in elastic beams: an intrinsic approach

The theory of non-uniform flexure and torsion of Saint-Venant's beam with arbitrary multiply connected cross section is revisited in a coordinate-free form to provide a computationally convenient context. Numerical implementations, by Matlab, are performed to evaluate the maximum elastic shear stresses in beams with rectangular cross sections for different Poisson's ratios. The deviations between the maximum and mean stresses are then diagrammed to adjust the results provided by Jourawski's method.